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The impossible barber and other bizarre thought experiments

How inventing riddles has revealed the flaws in our grasp of reality

Dmytro Zinkevych/Alamy Stock Photo

By Stephen Battersby

If you imagined that thought experiments were mere mental gymnastics meant to bamboozle the uninitiated, think again. Take Schrödinger’s cat, perhaps the most famous example, which involves a cat that is simultaneously alive and dead. It seems bizarre – and that’s the point. It was designed as a slap on the wrist for quantum theorists, to show that a theory that predicts such nonsense must be missing something. Current thinking is that perhaps nothing is missing, and quantum theory really is as weird as it seems.

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Here are seven classic thought experiments that might make you think…

The impossible barber

A certain barber is very particular about his work. He shaves every person who does not shave themselves, and no one who does shave themselves. So: does the barber shave himself? It doesn’t take long to see the contradiction: If he does, he can’t; if he doesn’t, he must. Such a barber can’t exist.

This barber is often used to illustrate a more abstract puzzle known as Russell’s paradox. In 1901, mathematician and philosopher Bertrand Russell was investigating set theory, a formal way of defining and dealing with collections of anything. At the time, one of its central ideas was that for every property you can define, there must be a set. There’s the set of all green things, and the set of all whole numbers except 4. You can also define sets of sets: say, the set of all sets that contain exactly two elements. The problem comes when pondering the possibility of a set of all sets that do not contain themselves — like the barber, this seems to be impossible.

The paradox exposed contradictions in much of the mathematics of the time, forcing Russell and others to try to devise more intricate logical footings for mathematics. Russell’s approach was to say that mathematical objects fall into a hierarchy of different “types”, each one built only from objects of lower type. Type theory has been used to design computer programming languages that reduce the chance of creating bugs. But it’s not the definitive solution – more than a century later, mathematicians are still arguing over the answer to Russell’s paradox.

Galileo’s balls

Galileo may never have dropped balls from the top of the leaning tower of Pisa, as the legend goes. But he did devise a simple thought experiment that told us something profound about gravity. Take two weights, one light, one heavy. If heavier objects fall faster than light ones, as Aristotle said, then the lighter weight will lag behind. That implies that when the two are tied together, they will fall more slowly than the heavy weight alone. But together, they weigh more than the heavy alone, so they should fall faster. Wait, so is it faster or slower?

As Galileo realised, acceleration due to gravity doesn’t depend on the mass of an object. This was a crucial result for the emerging science of physics, and Isaac Newton’s ideas of motion and universal gravitation. It even holds a germ of Einstein’s subtle theory of gravity. His general theory of relativity is rooted in the equivalence principle, the idea that gravity and acceleration are essentially the same thing— as Galileo glimpsed back in the 17th century.

Newton’s cannon

Take one gigantic cannon, put it on top of a mountain so high it reaches above the atmosphere, and fire horizontally. Irresponsible, perhaps, but instructive. If the cannonball is fired at a low speed, gravity will soon drag it down to the ground along a tightly curving arc. If you add more gunpowder, the ball will go faster and its arc will be more gradual, taking it further around the curve of the Earth. Fire it fast enough and the cannonball’s path will not meet the ground at all – it will fly all the way around and hit you in the back of the head. Here, have a go.

This thought experiment helped Newton show that gravity is a universal force: the force we see pulling cannonballs and apples to Earth can also explain the orbit of the moon around Earth, and Earth around the sun.

We are used to the idea of universal forces now. We know that nuclear reactions fuel the distant stars, and that exoplanets can be magnetic. But before Newton there was no expectation that the celestial realm should have the same rules as Earth. His cannonball blew a big hole in such heavenly pretentions.

Achilles and the tortoise

Two-and-a-half millennia ago, the Greek philosopher Zeno of Elea apparently proved that motion is an illusion. One of his paradoxes sets fleet-of-foot Achilles to chase a tortoise that has a small head start. Achilles can never catch the tortoise, argued Zeno, because first he must reach the point where the tortoise started, but by then the tortoise has moved on to a new position. So then Achilles must run there – by which time the tortoise has moved on again. The “dichotomy paradox” is more general: to cover any distance, you must first cover half that distance, then half of what’s left, then half of what’s left, and so on for ever. It seems that you can never get there, no matter what the original distance or how fast you move.

Since then, mathematicians have pointed out that although these arguments take an infinite amount of time to pan out, real motion doesn’t have to. We know for instance that that an infinite series of terms can add up to something finite. If you add an infinite series of fractions starting with ½ and halving in value with each new term (½ + ¼ +1/8…), the infinite sum is equal to 1. You can use maths like this to represent the distance travelled or the time taken in Zeno’s paradox, so — phew – motion is possible after all. That said, Zeno’s paradox may manifest itself for real in the quantum world.

The Chinese room

Can a computer be conscious? In an attempt to disprove this idea of “strong artificial intelligence”, John Searle, a philosopher at the University of California, Berkeley, imagined himself inside a room of dictionaries and rule books that hold instructions for translating Chinese to English and vice versa. Someone posts a question through the door written in Chinese, and using his rule books Searle works out an appropriate answer. To the questioner it would seem there is a mind in the room that understands Chinese, even though there isn’t. Searle claims that a hypothetical rule-bound computer designed to speak Chinese would be the same — a mere machine with no understanding.

There are many objections to this thought experiment. Some argue that although Searle does not understand Chinese, he is part of a larger system, including the rule books, that does. You might baulk at the idea that a mind could be made from a person plus some books, but it’s only a very dim mind, taking perhaps years or millennia to respond to one question.

Another interpretation is that Searle’s idea merely highlights the mystery of “other minds”: that you can’t know whether a computer, a penguin or the person next door is conscious in the same way as you are. If the Chinese room doesn’t disprove strong AI, thinking about it could help us to find out what’s missing from our understanding of consciousness.

Ride a light beam

In his Autobiographical Notes, Albert Einstein tells us how as a 16-year-old he imagined riding along with a light beam. If you could keep pace with it, the light must appear stationary, he imagined. Its oscillating electric and magnetic fields would be frozen. But that seems impossible. The equations developed by James Clerk Maxwell that describe the oscillations of electromagnetic fields forbid it, and we’ve certainly never seen such a thing as frozen light.

“One sees in this paradox the germ of the special relativity theory is already contained,” he wrote in 1947. As Einstein came to realise, the motion of light is the same no matter how fast you are moving. Even if you were travelling at almost the speed of light, the ray would still zip away from you at the same constant speed. This idea eventually led Einstein to an entirely new way of seeing the universe through the equations of special relativity, with their extraordinary predictions that time is elastic and that inert matter holds vast quantities of energy.

Laplace’s demon

Imagine a being that knows the place and motion of every particle in the universe. It also knows physics, and its mind works so fast that it can calculate how these particles will exert forces on one another, changing their motions. Can this intellect, described by Pierre Laplace in 1814, see the future of everything?

“Laplace’s demon”, as it became known, probes the idea of determinism. In a purely classical world, the demon seems to work. Chaos theory means that the future is ultra-sensitive to the past, but if the demon’s knowledge is infinitely precise, it could still know the fate of all.

Quantum mechanics may slay the demon. In mainstream quantum theory, events do not always have causes: radioactive decay and other things can happen spontaneously. But not all interpretations of quantum mechanics include this indeterminacy.

Even if the demon can live on in a universe governed by quantum mechanics, however, it probably doesn’t live around here. There is a mathematical argument that shows “the entire physical Universe cannot be fully understood by any single inference system that exists within it”. You might conceive of the demon as some kind of outside observer, but that opens another philosophical can of worms: is it meaningful to say that something can know all about our universe without having any physical effect on us?